The child who owns Kit never mentions his/her name, and it isn't shown when looked at either. It only shows after the animal has died.

If a child is allowed to keep Kit, one can't get the fox to leave. The only way to get rid of the pet is to kill it, which will result in the adopted children hating the Dragonborn. An easy way to avoid this is to cast Fury or Frenzy on Kit and have a follower, the child or any other NPC kill it.